if tan theta + sin theta = m and tan theta - sin theta = n evaluate m²- n²/√mn
Answers
Step-by-step explanation:
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Step-by-step explanation:
Given, m=tanθ+sinθ,n=tanθ−sinθ
We need to show m
2
−n
2
=4
mn
Taking L.H.S.,
m
2
−n
2
=(tanθ+sinθ)
2
−(tanθ−sinθ)
2
=tan
2
θ+sin
2
θ+2tanθ.sinθ−tan
2
θ−sin
2
θ+2tanθ.sinθ
=4tanθ.sinθ
Now, taking R.H.S.,
4
mn
=4
(tanθ+sinθ)(tanθ−sinθ)
=4
tan
2
θ−sin
2
θ
=4
cos
2
θ
sin
2
θ
−sin
2
θ
=4
cos
2
θ
sin
2
θ(1−cos
2
θ)
=4
sin
2
θ.tan
2
θ
=4tanθ.sinθ
Therefore, L.H.S. = R.H.S.